NUMERICAL METHOD FOR SOLVING THE EULER EQUATION FOR UNSTEADY TRANSONIC FLOWS OVER OSCILLATING AIRFOILS

Through transformations, the time-dependent boundary condition on the airfoil contour and the boundary condition at infinity are brought fixed to the boundaries of a finite domain. The boundary conditions can thus be satisfied exactly without increasing the computational time. The novel scheme is useful for computing transonic, strong disturbance, unsteady flows with high reduced frequencies. The scheme makes use of curvefitted orthogonal meshes and the lattice control technique to obtain the optimal grid distribution. The numerical results are satisfactory.

UNCONDITIONAL STABLE SOLUTIONS OF THE EULER EQUATIONS FOR TWO－AND THREE－D WINGS IN ARBITRARY MOTION

The work presented here shows the unsteady inviscid results obtained for the twoand three-dimensional wings which are in rigid and flexible osciliations.The results are generated by a finite volume Euler method. It is based on theRunge- Kutta time stepping scheme developed by Jameson et al.. To increase the timestep which is limited by the stability of Runge-Kutta scheme, the implicit residualsmoothing which is modified by using variable coefficients io prerent the loss of flowphysics for the unsteady flows is engaged in the calculations. With this unconditionalstable solver the unsteady flws about the wings in arbitrary motion can be receivedefficiently.The two- and three-dimensional rectangular wings which are in rigid andflexible pitching oscillations in the transonic flow are invesigated here, some of thecomputational results are compared with the experimental data. The influence of thereduced frequency for the two kinds of the wings are researched. All the results givenin this work are reasonable.

HIGH-ORDER ACCURATE AND HIGH-RESOLUTION UPWIND FINITE VOLUME SCHEME FOR SOLVING EULER/REYNOLDS-AVERAGED NAVIER-STOKES EQUATIONS

A high-order accurate explicit scheme is proposed for solving Euler/Reynolds-averaged Navier-Stokes equations for steady and unsteady flows, respectively. Baldwin-Lomax turbulence model is utilized to obtain the turbulent viscosity. For the explicit scheme, the Runge-Kutta time-stepping methods of third orders are used in time integration, and space discretization for the right-hand side (RHS) terms of semi-discrete equations is performed by third-order ENN scheme for inviscid terms and fourth-order compact difference for viscous terms. Numerical experiments suggest that the present scheme not only has a fairly rapid convergence rate, but also can generate a highly resolved approximation to numerical solution, even to unsteady problem.

Unsteady Boundary Layer Flow past a Stretching Plate and Heat Transfer with Variable Thermal Conductivity

An unsteady boundary layer flow of viscous incompressible fluid over a stretching plate has been considered to solve heat flow problem with variable thermal conductivity. First, using similarity transformation, the velocity components have been obtained, and then the heat flow problem has been attempted in the following two ways: 1) prescribed stretching surface temperature (PST), and 2) prescribed stretching surface heat flux (PHF) Flow and temperature fields have been analyzed through graphs. The expressions for skin friction and coefficient of convective heat transfer Nusselt number in PST and PHF cases have been derived.

TIME-PERIODIC ISENTROPIC SUPERSONIC EULER FLOWS IN ONE-DIMENSIONAL DUCTS DRIVING BY PERIODIC BOUNDARY CONDITIONS

We show existence of time-periodic supersonic solutions in a finite interval, after certain start-up time depending on the length of the interval, to the one space-dimensional isentropic compressible Euler equations, subjected to periodic boundary conditions. Both classical solutions and weak entropy solutions, as well as high-frequency limiting behavior are considered. The proofs depend on the theory of Cauchy problems of genuinely nonlinear hyperbolic systems of conservation laws.

非定常Euler方程数值计算中高精度格式应用

采用五阶精度加权紧致非线性格式(WCNS)和非定常"双时间步"方法求解非定常Euler方程,模拟NACA0012翼型强迫俯仰振动流场,研究了高精度格式应用到非定常计算时"双时间步"方法物理时间步长、子迭代收敛判据、子迭代步数以及物理时间导数离散方法对计算精度和计算效率的影响。

用 Euler 方程计算某些特殊的二维分离流动

采用非定常Ｅｕｌｅｒ方程、总体时间步长、隐式时间推进，计算了几种特殊类型钝体的二维有分离绕流。一类是光滑的圆柱体，在跨音速时计算出因激波诱导分离而周期性地脱落在尾迹上的非对称涡排。另一类是带尖角的物体，如三角形柱体，计算出从尖角处脱落在尾迹上的分离旋涡，形成卡门涡街。对这两类物体，计算的平均阻力系数，分离涡脱落的Ｓｔｒｏｕｈａｌ数以及流态都与实验接近。结果表明，对于这类绕流问题，用非定常Ｅｕｌｅｒ方程计算，具有一定工程实用意义。

跨音速非定常流动的Euler方程隐式解

在随时间变化的贴本坐标系中给出求解非定常Euler方程的连续通量分裂法。在此基础上建立了可用于跨音速非定常流动的Euler方程隐式求解法。采用特征向量变换,可在保证原方程组离散化精度的条件下使计算大为简化。针对振动翼绕流特点建立了固连于物体的动坐标与固定坐标间的关系。数值计算在动坐标中进行,既简化网格生成又保证在物面上满足边晃条件。对NACA64A-10冀型绕1/4弦点做简谐俯仰振动的非定常气动力进行了计算,给出了与实验结果基本相符的计算结果。此外,还给出翼型做沉浮及同时进行沉浮与俯仰二自由度振动的非定常气动力的计算结果。

翼-身-尾组合体的网格生成及 Euler 方程数值模拟

采用代数方法快速生成翼－身－尾组合体三维Ｏ－Ｃ型贴体网格。该代数网格生成方法使用双边界法，可保证物面附近近似正交，真实地模拟了机身头部及与机翼、尾翼交界处的形状，提高了代数方法生成网格的质量。并用中心格式有限体积法求解跨声速Ｅｕｌｅｒ方程，以某翼－身－尾组合体模型为例，计算结果与实验符合良好。

Hybrid Cartesian Grid Method for Moving Boundary Problems

A hybrid Cartesian structured grid method is proposed for solving moving boundary unsteady problems.The near body region is discretized by using the body-fitted structured grids,while the remaining computational domain is tessellated with the generated Cartesian grids.As the body moves,the structured grids move with the body and the outer boundaries of inside grids are used to generate new holes in the outside adaptive Cartesian grid to facilitate data communication.By using the alternating digital tree(ADT)algorithm,the computational time of hole-cutting and identification of donor cells can be reduced significantly.A compressible solver for unsteady flow problems is developed.A cell-centered,second-order accurate finite volume method is employed in spatial discretization and an implicit dual-time stepping low-upper symmetric Gauss-Seidel(LU-SGS)approach is employed in temporal discretization.Geometrybased adaptation is used during unsteady simulation time steps when boundary moves and the flow solution is interpolated from the old Cartesian grids to the new one with inverse distance weighting interpolation formula.Both laminar and turbulent unsteady cases are tested to demonstrate the accuracy and efficiency of the proposed method.Then,a 2-D store separation problem is simulated.The result shows that the hybrid Cartesian grid method can handle the unsteady flow problems involving large-scale moving boundaries.

Euler Solutions Using Approximate Boundary Conditions for Thin Airfoil with Small Oscillations

This paper presents an efficient numerical method for solving the unsteady Euler equations on stationary Cartesian grids. Wall boundary conditions are implemented on non moving mean wall positions by assuming the airfoil being thin and undergoing small deformation, but the mean angle of attack of the body can still be large and we use the full nonlinear Euler equation in the field for accurate resolution of shock waves and vorticity. The method does not require the generation of moving body fitted grids and thus can be easily deployed in any fluid structure interaction problem involving relatively small deformation of a thin body. We use the first order wall boundary conditions in solving the full Euler equation. Unsteady transonic flow is calculated about an oscillating NACA 0012 airfoil at free stream Mach number M ∞ =0.755, mean angle of attack α m =0.016, amplitude of pitching oscillation α 0 =2.51, reduced frequency κ = 0.081 4. The computed results, including surface pressure distribution, instantaneous lift and moment coefficients are compared with known experimental data. It is shown that the first order boundary conditions are satisfactory for airfoils of typical thicknesses with small deformation for unsteady calculations.

A Study of 2D Separated Cascade Flow at Large Angles of Attack Using Euler Equations

This paper presents a numerical simulation method developed for separated flow in cascades using Euler equations and demonstrates the Feasibility of this method MacCOrmack's two-steps explicit finite difference scheme is used to discretize the equations in conservation form,and the artificial viscosity is added to the discretized inviscid equations by means of the self-adapted filter technique.The initial separation boundary is given according to simple experimental results.The numerical simulation results including subsonic and transonic turbine cascades flow with of without separation show that the fundamental idea of this numerical method is reasonable and simple.The present stude indicates that for solving certain engineering problems it is a simple and effective tool for adding some viscosity corrections to inviscld flow model,especially the current when the Navier-Stokes equations have not been solved very effectively for verious complicated flows in turbonachinery.

ASecond-Order Finite-Difference Method for Compressible Fluids in Domains with Moving Boundaries

In this paper,we describe how to construct a finite-difference shockcapturing method for the numerical solution of the Euler equation of gas dynamics on arbitrary two-dimensional domainΩ,possibly with moving boundary.The boundaries of the domain are assumed to be changing due to the movement of solid objects/obstacles/walls.Although the motion of the boundary could be coupled with the fluid,all of the numerical tests are performed assuming that such a motion is prescribed and independent of the fluid flow.The method is based on discretizing the equation on a regular Cartesian grid in a rectangular domainΩ_(R)⊃Ω.Ωe identify inner and ghost points.The inner points are the grid points located insideΩ,while the ghost points are the grid points that are outsideΩbut have at least one neighbor insideΩ.The evolution equations for inner points data are obtained from the discretization of the governing equation,while the data at the ghost points are obtained by a suitable extrapolation of the primitive variables(density,velocities and pressure).Particular care is devoted to a proper description of the boundary conditions for both fixed and time dependent domains.Several numerical experiments are conducted to illustrate the validity of themethod.Ωe demonstrate that the second order of accuracy is numerically assessed on genuinely two-dimensional problems.

非结构动网格在三维可动边界问题中的应用

研究用于非结构动网格的弹簧近似方法,采用顶点弹簧描述,导出并讨论了弹簧倔强系数的取值.通过引入边界修正和扭转效应修正,对标准弹簧近似方法进行了改进.转动翼型算例的结果表明,改进后的方法大大提高了网格变形能力和网格质量.应用该动网格方法耦合求解基于(Arbitrary Lagrangian-Eulerian,ALE)描述的三维Euler方程,模拟了作俯仰振动的矩形机翼绕流,计算结果与实验数据及文献计算结果十分一致.作为多个自由刚体与流体耦合运动问题的简单例证,耦合刚体动力学方程,模拟了激波与双立方体的相互作用,得到了非定常流场结构.研究表明,基于弹簧近似的非结构动网格与有限体积格式流场解算器相结合,是模拟包含运动边界的非定常流动问题的有效方法.

用动态非结构重叠网格法模拟三维多体相对运动绕流

发展了动态非结构重叠网格技术。提出了基于阵面推进的相邻单元搜索算法和适合于高阶空间离散格式的网格间插值关系定义方法,前者提高了非结构重叠网格在模拟多体间具有大幅相对运动的非定常问题时网格处理的效率,而后者保证了网格间边界附近流场求解精度。非定常Euler方程的求解采用有限体积法,通量计算采用Osher格式,时间积分采用双时间步长推进。为验证本文提出的方法,分别对定常M6机翼绕流和机翼下外挂物投放过程的非定常绕流进行了模拟,结果表明该方法提高了非结构重叠网格在模拟多体间具有大幅相对运动问题时的效率,保证了网格边界处的精度,具有较高的工程应用价值。

跨音速颤振计算方法研究

采用快捷的动态弹性变形技术生成三维块结构动态贴体网格。结构运动方程采用二级精度的龙格—库塔时间推进。气动力求解则用非定常欧拉方程的双时间有限体积推进,外时间为物理时间,与结构运动方程同步;对每个真实物理时间步采用五步Runge Kutta时间推进进行虚拟定常迭代。针对可压流颤振计算中存在的质量不相似问题,采用变质量、变刚度的方法计算出质量匹配点处的颤振速压值,并根据颤振速压随质量或刚度倍数的变化趋势,得出可用的飞机跨音速颤振速压。对一飞机简化外形的跨音速颤振特性计算得到了合理的计算结果。

用边界元法分析油藏不稳定渗流问题

油藏几何形状对井底压力动态响应有显著的影响。常规的解析方法却只能解决一些简单、规则形状的问题。对于非规则形状的问题,边界元法是有效手段之一。它具有降维、计算速度快、精度高以及求解内点方便等特点。根据边界元理论推导了任意形状油藏不稳定渗流的计算方法,并将其计算结果与常规解析方法求解结果相比较。边界元法在油藏数值模拟和试井领域有广阔的应用前景。

高超声速验证飞行器助推分离段流场数值研究

为了模拟有相对运动的多体非定常绕流 ,发展了基于弹簧近似的非结构动网格方法及耦合动网格的Euler解算器。作为方法验证 ,模拟了俯仰振动的NACA0 0 12翼型绕流 ,计算结果与实验及文献结果非常接近。采用二维近似外形 ,对高超声速验证飞行器助推分离段流场进行了数值研究 ,得到了不同时刻由于多体相对运动形成的干扰流场结构以及分离过程的气动力参数。

安溢活门动态特性的流固耦合数值分析

采用流固耦合数值模拟技术研究安溢活门/气罐系统充放气动态特性。计算中所用流体控制方程为三维Euler方程,采用非结构网格有限体积格式求解;固体结构的运动方程由拉格朗日方程导出,采用纽马克方法求解。提出可变边条零厚度虚拟挡板技术,模拟主活塞由闭到开的瞬态过程。计算得到开阀和关阀过程中弹性元件的动态响应特性,结果表明,主膜片在行程上限和下限处均产生高频振荡运动,在开阀和关阀瞬间主膜片受到冲击载荷作用,并且关阀时载荷出现振荡。这种振荡冲击可能是导致活门鸣叫和膜片破坏的原因。

动边界问题无网格算法及其动态点云

提出基于Delaunay图映射的点云运动处理技术。该技术使点云随物体运动再生成问题变成简单的映射关系,无需迭代,且能保持点云结构不变,故有效地避免了因点云结构变动而产生的附加计算量。运用该技术,结合非定常Euler方程双时间步无网格算法,成功地模拟了翼型的跨声速非定常流动。其中物理时间迭代采用二阶隐式格式,伪时间迭代采用四步龙格-库塔显式格式。文中给出了不同翼型的计算结果,并与实验结果进行了比较,取得了令人满意的结果。